Collective Magnetic Excitations in the Spin Ladder ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathbf{O}}_{41}$ Measured Using High-Resolution Resonant Inelastic X-Ray Scattering

We investigate magnetic excitations in the spin-ladder compound ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ using high-resolution Cu $L_3$ edge resonant inelastic x-ray scattering (RIXS). Our findings demonstrate that RIXS couples to two-triplon collective excitations. In contrast to inelastic neutron scattering, the RIXS cross section changes only moderately over the entire Brillouin zone, revealing high sensitivity also at small momentum transfers, allowing determination of the two-triplon energy gap as $100\pm 30\mathrm{meV}$. Our results are backed by calculations within an effective Hubbard model for a finite-size cluster, and confirm that optical selection rules are obeyed for excitations from this spherically symmetric quantum spin-liquid ground state.

Figure 1

Cu $L_3$ XAS and RIXS data from ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$: (a) XAS (left) measured in total fluorescence yield (TFY) mode at 15 K; Right: sketch of the experimental setup and definition of symbols: $\mathit{k}$ (${\mathit{k}}^{\prime }$) wave vector of the incident (scattered) light, $\theta$ incidence angle, $\Psi$ scattering angle, $\mathit{q}={\mathit{k}}^{\prime }-\mathit{k}$ transferred momentum; (b) RIXS for energy $C$—at 15 K (solid line) and room temperature (RT) (dashed line)—and off-resonance; (c) RIXS energy series ($A\mathrm{\text{−}}D$) at 15 K. All data were measured at $\Psi =90°$, $\theta =20°$ using $\sigma$-polarized light.

Figure 2

Dispersion of magnetic excitations in Cu $L_3$ RIXS: black (blue) corresponds to $\Psi =90°$ (130°).

Figure 3

(a) RIXS data as intensity map vs momentum and energy transfer after subtraction of the elastic signal. We overlay the extracted dispersion curve of the lower boundary of the two-triplon continuum with open black diamond symbols. For comparison we plot calculated one- (red dashed line) and two-triplon (black full lines) dispersion curves from $K$. Schmidt et al. [22]. The right axis is scaled in units of $J$ along the rungs ($J_{\perp }$), extracted from the theoretical model below. (b) RIXS spectra measured close to the BZ center point for $q_c=-0.04\times 2\pi /c_L$, using $\sigma$- and $\pi$-polarized light. The $\pi$-polarized spectrum is fitted by two Gaussians, the residual is represented by the thin blue line.

Figure 4

Cu $L_3$ RIXS simulation for an effective Hubbard model on an eight-site ladder cluster.